When the positive integer \(x\) is divided by the positive integer \(y,\) the quotient is \(3\) and the remainder is \(z

[VJesus12]

When the positive integer \(x\) is divided by the positive integer \(y,\) the quotient is \(3\) and the remainder is \(z.\) When \(z\) is divided by \(y,\) the remainder is \(2.\) Which of the following could be the value of \(x?\)

I. \(5\)
II. \(8\)
III. \(32\)

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Answer: C

Source: GMAT Club Tests

[swerve]

When the positive integer \(x\) is divided by the positive integer \(y,\) the quotient is \(3\) and the remainder is \(z.\) When \(z\) is divided by \(y,\) the remainder is \(2.\) Which of the following could be the value of \(x?\)

I. \(5\)
II. \(8\)
III. \(32\)

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Answer: C

Source: GMAT Club Tests

Here we have two equations,

\(\dfrac{x}{y}=3+z\quad\) and \(\quad \dfrac{z}{y} = k+2\)

Now, let's start plugging in the options -

I. 5 - No possible satisfying conditions can be found out.
II. 8 - No possible satisfying conditions can be found out.
III. 32 - Let's try

\(\dfrac{32}{y}=3+z \quad \Longrightarrow \quad \dfrac{32}{9}=3(\text{Quotient})+5(\text{Remainder})\)

Let's try the second step now

\(\dfrac{z}{y}=k+2\)

\(\dfrac{5}{y}=k+2\)

\(\dfrac{5}{3}=1(\text{Quotient})+2(\text{Remainder})\)

Therefore, C